Hard factoring problem

Hi. I had this practice problem for learning factoring:

$\displaystyle 4a^2c^2-(a^2-b^2+c^2)^2$

I eventually gave up, but the answer was:

$\displaystyle (a+b+c)(a+b-c)(a-b+c)(b-a+c)$

I'm wondering if anyone understands what steps were taken. (Which unfortunately were not described on the answer page.) I tried distributing, grouping, and factoring formulas, but the closest I got (if you want to call it progress) was:

$\displaystyle (a^2+b^2)^2+c(3a^2c-a^2+b^2+b^2c-c^3)$

Anyway, I'm about at the point now where I can hardly see the numbers clearly...